Is there any way to select all the "edge faces" of this cut-out sphere?
I suggest using select similar > perimeter. Press  Shift  + G to bring up the selection menu after selecting a face loop similar to the desired loop with  Alt  +  RMB , then adjust the threshold until you have the desired results. Double check your model to make sure there isn't a missed face anywhere, but this should get all of them. In my example, the threshold was 0.030